The recent two-dimensional (2-D) parametric formulation of the high fidelity generalized method of cells (HFGMC) reported by the authors is generalized for the micromechanical analysis of three-dimensional (3-D) multiphase composites with periodic microstructure. Arbitrary hexahedral subcell geometry is developed to discretize a triply periodic repeating unit-cell (RUC). Linear parametric-geometric mapping is employed to transform the arbitrary hexahedral subcell shapes from the physical space to an auxiliary orthogonal shape, where a complete quadratic displacement expansion is performed. Previously in the 2-D case, additional three equations are needed in the form of average moments of equilibrium as a result of the inclusion of the bilinear terms. However, the present 3-D parametric HFGMC formulation eliminates the need for such additional equations. This is achieved by expressing the coefficients of the full quadratic polynomial expansion of the subcell in terms of the side or face average-displacement vectors. The 2-D parametric and orthogonal HFGMC are special cases of the present 3-D formulation. The continuity of displacements and tractions, as well as the equilibrium equations, are imposed in the average (integral) sense as in the original HFGMC formulation. Each of the six sides (faces) of a subcell has an independent average displacement micro-variable vector which forms an energy-conjugate pair with the transformed average-traction vector. This allows generating symmetric stiffness matrices along with internal resisting vectors for the subcells which enhances the computational efficiency. The established new parametric 3-D HFGMC equations are formulated and solution implementations are addressed. Several applications for triply periodic 3-D composites are presented to demonstrate the general capability and varsity of the present parametric HFGMC method for refined micromechanical analysis by generating the spatial distributions of local stress fields. These applications include triply periodic composites with inclusions in the form of a cavity, spherical inclusion, ellipsoidal inclusion, discontinuous aligned short fiber. A 3-D repeating unit-cell for foam material composite is simulated.